Reaction Limited Regime-Uniqueness

The index of a steady state T r) is calculated from the eigenvalue problem 1 d /, d AH,  

the inequality (2.6.19) is a sufficient condition for the uniqueness of the steady state. A simple result of (2.6.19) is that the steady state is unique for a sufficiently dilute gas mixture (small c q) [30]. 

As in the case of a single reaction, Section 2.3, we shall show that when the reaction rates are small compared to the transport rates the steady state is unique. In this respect, the situation is analogous with that of the uniform open systems of Section 1.9. 

The proof will make use of Theorem (2.6.2). By forming the scalar product of Eq. (2.6.5) with integrating, and integrating by 

Since only real X are of interest, the denominator can be estimated by considering real only  

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The remaining four sections will be concerned with problems of uniqueness, stability, and asymptotic behavior of the steady states. The results are in close analogy with those obtained in Section 1.9, especially in regard to the behavior of the system in the transport and reaction limited regimes. 

In the classical free statistics, the number of the functional groups on the surface of a tree-like cluster is of the same order of that of the groups inside the cluster, so that a simple thermodynamic limit without surface term is impossible to take. The equilibrium statistical mechanics for the polycondensation was refined by Yan [14] to treat surface correction in such finite systems. He found the same result as Ziff and Stell. Thus the treatment of the postgel regime is not unique. The rigorous treatment of the problem requires at least one additional parameter defining relative probability of occurrence of infra- and intermolecular reactions in the gel. 

Cluster impact-induced chemistry (CIC, [1-3]) has the characteristics of both isolated elementary processes as in gas phase dynamics and of reactions in solution. Yet it is not a bridge between the two limits but a quite distinct regime in dynamics which provides for many unique possibilities and for a degree of control not readily available otherwise. Here we discuss CIC as studied on the computer. The present result only show what can happen. It remains for our colleagues to tell us what actually does happen. 

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